```Date: Feb 2, 2013 5:26 AM
Author: mina_world
Subject: Quadruple with (a, b, c, d)

Hello teacher~Suppose not all four integers a, b, c, d are equal.Start with (a, b, c, d) and repeatedly replace (a, b, c, d)by (a-b, b-c, c-d, d-a).Then at least one number of the quadruple will eventuallybecome arbitrarily large.-----------------------------------------------------------------------Solution)Let P_n = (a_n, b_n, c_n, d_n) be the quadruple after n iterations.Then a_n + b_n + c_n + d_n = 0 for n >= 1.A very importand function for the point P_n in 4-spaceis the square of its distance from the origin (0,0,0,0),which is (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2.....omissionFor reference, text copy with jpg.http://board-2.blueweb.co.kr/user/math565/data/math/olilim.jpg...for n >=2,(a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2>= {2^(n-1)}*{(a_1)^2 + (b_1)^2 + (c_1)^2 + (d_1)^2}.The distance of the points P_n from the origin increases without bound,which means that at least one component must become arbitrarily large----------------------------------------------------------------Hm, my question is...I know that  a_n + b_n + c_n + d_n = 0andI know that (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2 increases without bound.I can't understand that "at least one component must become arbitrarily large".I need your logical explanation or proof.
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